As discussed earlier, our result is a middle ground between the non-convergence (in norm topology) and the pointwise asymptotic normality of the FPCA estimator. The key contribution of our result is to further understand the asymptotic distributional properties of the FPCA estimator. Up to our knowledge, only a few select studies (most notably [1]}) have studied this problem from the perspective of inference. There are more, but still few, studies of the asymptotic distributional properties of the FPCA estimator for the purpose of prediction; for instance, see [2]} and [3]}, among others.
